What is Density function: Definition and 192 Discussions

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.
In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to 1.
The terms "probability distribution function" and "probability function" have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion. In general though, the PMF is used in the context of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.

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  1. P

    I Density fluctuations and the color of the sky

    From posts like this I get that the color of the sky is explained by Rayleigh scattering, but needs density fluctuations. However as atoms are not uniform and are localized, the density is $$\rho(\mathbf r)=\sum_i \delta(\mathbf r-\mathbf r_i)$$ where ##i## sums over all the atom positions...
  2. chwala

    Solve this problem that involves a probability density function

    I am refreshing on this; ..after a long time... Note that i do not have the solution to this problem. I will start with part (a). ##f(u)= 3u-\dfrac{3u^2}{2k}## with limits ##0≤u≤k## it follows that, ##3k - \dfrac{3k}{2}=1## ##\dfrac{3k}{2}=1## ##k=\dfrac {2}{3}## For part (b)...
  3. chwala

    Solve the problem involving probability density function

    This is the question: This is the ms solution- from Further Maths paper. My question is referenced to the highlighted part. I can see they substituted for the lower limit i.e ##x=1## to get: ##F(x)=\dfrac{x^3-1}{63}## supposing our limits were; ##2≤x≤4## would the same approach apply? Anything...
  4. J

    Using Gauss's Law to Calculate Charge Density Function

    I've attached what I have so far. Used Gauss's law, everything seemed to make sense except the units don't work out in the end. The charge density function if given by: r(z)=az, where z is the perpendicular distance inside the plane.
  5. chwala

    Problem involving Probability density function

    I just want to be certain, i think the inequality indicated is not correct...ought to be less than. Kindly confirm...This is a textbook literature.
  6. A

    A Solving an Integral involving a probability density function

    In an article written by Richard Rollleigh, published in 2010 entitled The Double Slit Experiment and Quantum Mechanics, he argues as follows: "For something to be predictable, it must be a consistent measurement result. The positions at which individual particles land on the screen are not...
  7. A

    I The probability density function for the double-slit experiment

    I am desperate. I've scoured the web for the formula for the probability density function for the interference pattern obtained in the double slit experiment with both slits open. So I want to know the probability density function and not the intensity function. I prefer not to have references...
  8. PainterGuy

    Random variable and probability density function

    Hi, I was trying to solve the attached problem which shows its solution as well. I cannot understand how and where they are getting the equations 3.69 and 3.69A from. Are they substituting the values of θ₁ and θ₂ into Expression 1 after performing the differentiation to get equations 3.70 and...
  9. N

    A Probability Density Function: Converting Experimental Observations to PDF

    Hi All I am currently doing Master in data science. I came across the function PDF probability density function which is used to find cumulative probability(range) of a continuous random variable. The PDF probability density function is plotted against probability density in y-axis and...
  10. P

    I Find P(X+Y>1/2) for given joint density function

    Hey everybody, :smile: I have a joint density of the random variables ##X## and ##Y## given and want to find out ##P(X+Y>1/2)##. The joint density is as follows: $$f_{XY}(x,y) = \begin{cases}\frac{1}{y}, &0<x<y,0<y<1 \\ 0, &else \end{cases}$$ To get a view of this I created a plot: As...
  11. M

    Finding marginal distribution of 2d of probability density function

    Hi, I have question about finding marginal distributions from 2d marginal pdfs that lead to the probabilities being greater than 1. Question: If we have the joint probability distribution ## f(x, y) = k \text{ for} |x| \leq 0.5 , |y| \leq 0.5 ## and 0 otherwise. I have tried to define a square...
  12. A

    A How to transform a probability density function?

    I have the following probability density function (in Maple notation): f (x) = (1 / ((3/2) * Pi)) * (sin (x)) ** 2 with support [0; 3 * Pi] Now I want to transform x so that 0 -> (3/2) * Pi and 3 * Pi -> (15/2) * Pi and the new function is still a probability density function. How should I...
  13. A

    A How to change the support of a probability density function?

    Given the support [a, b] of a probability density function. How can I change the formula for the probability density function with a support [u, v]? Example: Given the beta distribution with support [a=0,b=1]: $$\frac{x^{p-1} (1-x)^{q-1}}{Beta(p,q)}$$ Then the beta distribution with support...
  14. A

    On the width of the kinetic energy distribution of a gas

    In these lecture notes about statistical mechanics, page ##10##, we can see the graph below. It represents the distribution (probability density function) of the kinetic energy ##E## (a random variable) of all the gas particles (i.e., ##E=\sum_{i}^{N} E_{i}##, where ##E_{i}## (also a random...
  15. megf

    I Probability Density Function of the Product of Independent Variables

    How do I find the probabilty density function of a variable y being y=ab, knowing the probabilty density functions of both a and b? I know how to use the method to calculate it for a/b - which gives 1/pi*(a²/b²+1) - using variable substitution and the jacobian matrix and determinant, but which...
  16. Vicol

    Probability density function in classical mechanics

    Probability density function plays fundamental role in qunatum mechanics. I wanted to ask if there is any analogous density function in classical mechanics. Obviously if we solve Hamilton equations we get fully deterministic trajectory. But it should be possible to find function which shows...
  17. shahbaznihal

    I Phase space density function and Probability density function

    I am reading a text which talks about the WIMP speed distribution in the galactic halo in the frame of the Sun and Earth. The point where I am stuck it is trying to explain the concept of Gravitational Focusing of WIMPs at the location of the Earth due to the gravitational well of the Sun...
  18. Robin04

    Density function, calculating the mass

    Homework Statement The density of a rod in function of space is given as ##\rho (x)=\frac{c}{x^2}## 1. What kind of density is this? 2. What is the dimension of ##c##? 3. What is the mass of the rod in the intervals - [1 m, 2 m], - (1 m, 2 m), - (0 m, 1 m), - [0 m, 1 m]? 4. Can a plate with...
  19. S

    B Expectation of probability density function

    E(X) of probability density function f(x) is \int x f(x) dx E(X2) of probability density function f(x) is \int x^2 f(x) dx Can I generalize it to E(g(x)) of probability density function f(x) = \int g(x). f(x) dx ? I tried to find E(5 + 10X) from pdf. I did two ways: 1. I found E(X) then using...
  20. CivilSigma

    Find the PDF in terms of another variable

    Homework Statement For $$f_x(x)=4x^3 ; 0 \leq x \leq 1$$ Find the PDF for $$ Y < y=x^2$$ The Attempt at a Solution So, we take the domain on x to be: $$0\leq x \leq \sqrt y$$ and integrate: $$ \int_0^{\sqrt y} f_x(x) dx = \int_0^{\sqrt y} 4x^3 dx$$ Do we integrate with respect to x or y...
  21. Another

    I Please help explain the probability density function

    ##E = \frac{1}{2}(kx^2+m \dot{x}^2)## ## \frac{2E - kx^2}{m}=\dot{x}^2## ##\frac{dx}{dt} = \sqrt{\frac{2E - kx^2}{m}}## or ## dt = \sqrt{\frac{m}{2E - kx^2}}dx ## ⇒##= \frac{1}{\sqrt{\frac{2E - kx^2}{m}}}dx## My Question please help me. 1. I know ##T = 2\pi\sqrt{\frac{m}{k}} .## but i don't...
  22. S

    MHB Given probability density function find its cumulative distribution function

    Hi :) Here's my problem along with what I've done. Here is the problem: That is the p.d.f. of a random variable X. I have to find the cdf. I don't know which I should do so I tried it two ways. First: $\int_{-1}^{1} \ \frac{2}{\pi(1+x^{2})} dx = {{\frac{2}{\pi} arctan(x)]}^{1}}_{-1}=1$...
  23. S

    B How can I model a density function of a compressible fluid?

    I have a cylinder of some dimensions. I have a compressible liquid inside. Assuming a constant temperature, no atmosphere, no convection currents within, because it is in a cylinder, there will be no variations in density horizontally (the fluid will have time to settle). Now because there is...
  24. B

    MHB Probability Density Function of Average Value of Log-Normal Trials

    X is a random variable that follows the Log-Normal probability density function. n indipendent trials are carried out. We want to know the probability density function of the random variable Y, that is defined as the average value of the “n” outcomes of the trials described above.
  25. K

    I Use of the density function in physics

    For continuos groups one introduces a density function for an invariant measure when summing over the group elements. I learned a little about these concepts in a pure mathematical book. I was thinking about their utility in physics.. I know they probably do.. What physical areas could these...
  26. P

    Finding Probability Density Functions for Independent Random Variables

    Homework Statement Hello! I'm trying to understand how to solve the following type of problems. 1) Random variables x and y are independent and uniformly distributed on the interval [0; a]. Find probability density function of a random variable z=x-y. 2) Exponentially distributed (p=exp(-x)...
  27. Mehmood_Yasir

    Probability and pedestrian wait time density function

    Homework Statement Pedestrians approach to a signal at the crossing in a Poisson manner with arrival rate ##\lambda## arrivals per minute. The first pedestrian arriving the signal starts a timer ##T## then waits for time ##T##. A light is flashed after time T, and all waiting pedestrians who...
  28. M

    Probability density function for a given problem

    https://www.quora.com/How-can-I-find-the-probability-density-function-of-a-continuous-random-variable-in-a-given-problem/answer/Maxime-Denis-2 How can I find the probability density function of a continuous random variable in a given problem? A mass m swings at the end of a rope (of length L)...
  29. T

    I Understanding the probability density function

    Hi all This is not a homework question but something work related which I am having difficulty understanding which I was hoping someone from the community could help me with. I am trying to understand how to interpret & create the probability density function plot from a set of data. For...
  30. M

    Finding k in a probability density function

    Homework Statement Let X, Y, and Z have the joint probability density function f(x,y,z) = kxy2z for 0 < x, y < 1, and 0 < z < 2 (it is defined to be 0 elsewhere). Find k. Homework Equations Not sure how to type this in bbcode but: Integrate f(x,y,z) = kxy2z over the ranges of x (zero to...
  31. T

    I What Does The Probability Density Function Tell You?

    Hello All I was wondering if someone could help explain what the probability density function tells you. I am trying to learn about surveying and the PDF keeps cropping up and I do not fully understand it. For example I have:- measured a single angle 15 times calculated my Standard Deviation...
  32. Saracen Rue

    Probability Density Function problem

    Homework Statement Presume the relation ##\frac{x}{x+y^2}-y=x## is defined over the domain ##[0,1]##. (a) Rearrange this relation for ##y## and define it as a function, ##f(x)##. (b) Function ##f(x)## is dilated by a factor of ##a## from the y-axis, transforming it into a probability density...
  33. Saracen Rue

    Probability Density Function problem

    Homework Statement Q6. A function, ##f\left(x\right)=\frac{ax+1}{\left(ax-1\right)^3-\frac{a}{\left(x-1\right)^2-1}}##, can be defined as a PDf over the domain ##(0, 2)##. Express answers to 4 decimal places unless specified otherwise; (a) Find the value of ##a## given that ##f(x)## is a PDf...
  34. Saracen Rue

    Difficult Probability Density Function Question

    Homework Statement A function, ##f\left(x\right)=\left|a^{\frac{\sin \left(x\right)}{\ln \left(ax\right)}}-\frac{x}{a}\right|##, intersects with another function, ##g\left(x\right)=\left|\frac{sin(ln(\sqrt{x}-\sqrt{a}))}{x^2-a^2}\right|##, at point ##Q(b,f(b))## and point ##R(c,f(c))##. A...
  35. NatFex

    I Sum of Probability Density Function > 1?

    I have a Stats exam on Wednesday and while I thought I was quite well-versed, I've gone back over to the very basics only to find myself confused at what should be introductory. Suppose I have a continuous random variable modeled by a probability density function: $$f(x)=2x$$ Obviously the...
  36. G

    Mass Given Density Function and Area

    Homework Statement A quarter disc of radius 3 cm lies in the first quadrant. The areal density is (1.2 g/cm3)x + (0.7 g/cm3)y. Determine the mass of this object. Homework Equations The Attempt at a Solution For my bounds: x: 0 to 3 y: 0 to Sqrt[3 - x^2] When I took this integral I got...
  37. Schwarzschild90

    Calculating Standard Deviation for a Probability Density Function?

    Homework Statement Homework Equations See below The Attempt at a Solution \begin{align} \begin{split} p(x) = C \ x \ exp(-x/ \lambda) \end{split} \end{align} If $p(x)$ is a probability density function on the interval $ 0 \textless x \textless + \infty $ , then it follows...
  38. Jan Wo

    Getting electron density from probability density function

    Hello Lastly I was thinking a lot about electron density definition. It is not intuitive for me and I'm looking for any mathematical tool that could explain it to me more. My friend told me about idea to derivate it from propability density function using Dirac delta distribution. I'd like to...
  39. T

    Integrating a theta function/2d probability density function

    Homework Statement A two-dimensional circular region of radius a has a gas of particles with uniform density all traveling at the same speed but with random directions. The wall of the chamber is suddenly taken away and the probability density of the gas cloud subsequently satisfies $$...
  40. X

    Determine charge at origin, based on charge density function

    Homework Statement a) and b) are no problem. I need help to solve c) and d) Homework Equations c) Delta dirac function Gauss' law d) Gauss' law ## \int_V {\rho \, d\tau} = Q_{enclosed} ## The Attempt at a Solution By taking laplace on the potential I get: ## \rho(\mathbf{r}) =...
  41. S

    Probability density function of simple Mass-Spring system.

    Homework Statement We know that after long run of simple mass-spring system, there should be a probability of finding the mass at certain points between -A and A.. Obviously in probability of finding the particle near A or -A is higher than finding the particle at 0, because the speed is the...
  42. diracdelta

    Probability density function of uniform distribution

    Homework Statement Random variable X is uniformly distributed on interval [0,1]: f(x)=\begin{cases} 1 & \text{ if } 0\leq x\leq 1\\ 0 & \text{ else} \end{cases} a) Find probability density function ρ(y) of random variable Y=\sqrt{X} +1 I tried like this. Is it good, if no why not...
  43. Lewis7879

    Joint problem density function problem

    I need help guys I can't understand this Can anyone explain thoroughly how do I form the range for this question? f(x,y)= e-x for 0≤x≤y≤∞ 0 Otherwise Find P(x+y≤1) I attempted this by integrating through the range of 0≤y≤(1-x) and 0≤x≤∞ but that...
  44. D

    What are the Properties of Joint Density Functions?

    Hello, Homework Statement The joint probability density function of X and Y is given by f(x,y)=c*1(|y|<x<1) 1 is the indicator function -Find c -Find the marginal densities of X and Y -Find the means, variances and the covariance -Find conditional densities, means and variances of X given...
  45. Calpalned

    Density function for a normal distribution

    Homework Statement I have to prove that ## \int e^{\frac{x^2}{-2}}dx ## from +∞ to -∞ = ##\sqrt{2\pi} ## Homework Equations N/A The Attempt at a Solution My GSI went from 1) ## \int e^{\frac{x^2}{-2}}dx ## from +∞ to -∞ = ##\sqrt{2\pi} ## to 2) ## (\int e^{\frac{x^2}{-2}}dx)(\int...
  46. C

    Probability Question - Joint density function

    Before I begin, here is the question: If the PDF of two independent random variables X and Y are: f(x) = exp(-x)u(x) f(y) = exp(-y)u(y) Determine the join probability density function (JPDF) of Z&W defined by: Z = X+Y W = X/(X+Y). So, I know how to solve this except for one thing. How do I get...
  47. Mr Davis 97

    What is a Probabilty Density Function?

    What really is a probability density function for continuous random variables? I know that the probability for a single value occurring in a continuous probability distribution is so infinitesimal that it is considered 0, which is why we use the cumulative distribution function that is the the...